Calculus Examples

$\arcsin (x) + \arcsin (y) = \frac{π}{2}$ , $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step 1

Subtract $\arcsin (y)$ from both sides of the equation.

$\arcsin (x) = \frac{π}{2} - \arcsin (y)$

Step 2

Rewrite the equation as $\frac{π}{2} - \arcsin (y) = \arcsin (x)$ .

$\frac{π}{2} - \arcsin (y) = \arcsin (x)$

Step 3

Subtract $\frac{π}{2}$ from both sides of the equation.

$- \arcsin (y) = \arcsin (x) - \frac{π}{2}$

Step 4

Divide each term in $- \arcsin (y) = \arcsin (x) - \frac{π}{2}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \arcsin (y) = \arcsin (x) - \frac{π}{2}$ by $- 1$ .

$\frac{- \arcsin (y)}{- 1} = \frac{\arcsin (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\arcsin (y)}{1} = \frac{\arcsin (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 4.2.2

Divide $\arcsin (y)$ by $1$ .

$\arcsin (y) = \frac{\arcsin (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\arcsin (x)}{- 1}$ .

$\arcsin (y) = - 1 \cdot \arcsin (x) + \frac{- \frac{π}{2}}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \arcsin (x)$ as $- \arcsin (x)$ .

$\arcsin (y) = - \arcsin (x) + \frac{- \frac{π}{2}}{- 1}$

Step 4.3.1.3

Dividing two negative values results in a positive value.

$\arcsin (y) = - \arcsin (x) + \frac{\frac{π}{2}}{1}$

Step 4.3.1.4

Divide $\frac{π}{2}$ by $1$ .

$\arcsin (y) = - \arcsin (x) + \frac{π}{2}$

Step 5

Take the inverse arcsine of both sides of the equation to extract $x$ from inside the arcsine.

$y = \sin (- \arcsin (x) + \frac{π}{2})$

Step 6

Rewrite the equation as $\sin (- \arcsin (x) + \frac{π}{2}) = y$ .

$\sin (- \arcsin (x) + \frac{π}{2}) = y$

Step 7

Take the inverse sine of both sides of the equation to extract $\arcsin (x)$ from inside the sine.

$- \arcsin (x) + \frac{π}{2} = \arcsin (y)$

Step 8

Subtract $\frac{π}{2}$ from both sides of the equation.

$- \arcsin (x) = \arcsin (y) - \frac{π}{2}$

Step 9

Divide each term in $- \arcsin (x) = \arcsin (y) - \frac{π}{2}$ by $- 1$ and simplify.

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Step 9.1

Divide each term in $- \arcsin (x) = \arcsin (y) - \frac{π}{2}$ by $- 1$ .

$\frac{- \arcsin (x)}{- 1} = \frac{\arcsin (y)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

$\frac{\arcsin (x)}{1} = \frac{\arcsin (y)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 9.2.2

Divide $\arcsin (x)$ by $1$ .

$\arcsin (x) = \frac{\arcsin (y)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Move the negative one from the denominator of $\frac{\arcsin (y)}{- 1}$ .

$\arcsin (x) = - 1 \cdot \arcsin (y) + \frac{- \frac{π}{2}}{- 1}$

Step 9.3.1.2

Rewrite $- 1 \cdot \arcsin (y)$ as $- \arcsin (y)$ .

$\arcsin (x) = - \arcsin (y) + \frac{- \frac{π}{2}}{- 1}$

Step 9.3.1.3

Dividing two negative values results in a positive value.

$\arcsin (x) = - \arcsin (y) + \frac{\frac{π}{2}}{1}$

Step 9.3.1.4

Divide $\frac{π}{2}$ by $1$ .

$\arcsin (x) = - \arcsin (y) + \frac{π}{2}$

Step 10

Take the inverse arcsine of both sides of the equation to extract $x$ from inside the arcsine.

$x = \sin (- \arcsin (y) + \frac{π}{2})$

Step 11

The equation can not be solved. The given interval accounts for only one variable, but $3$ are present in the equation $x = \sin (- \arcsin (y) + \frac{π}{2})$ .

No solution